Method of creating curved surface

ABSTRACT

A method of creating the curved surface of a body including steps of generating a plurality of intermediate sections and finding a section curve in each of the intermediate sections, in accordance with predetermined rules, from section data specifying given sections of the body and from data specifying sections curves in respective ones of the given sections, and generating the curved surface of the body from a collection of intermediate section curves in the general intermediate sections.

BACKGROUND OF THE INVENTION

This invention relates to a method of creating a curved surface of athree-dimensional body, and more particularly to a method of the typedescribed that is ideal for the preparation of a numerical control taperequired for the numerically controlled machining of a three-dimensionalmetal mold or the like.

A curved surface of a three-dimensional metal mold or the like, whendrawn out on the plane of a blueprint is generally represented by aplurality of given section curves, but no data is shown for the shapesof the areas lying between the adjacent given section curves. Whenmachining the workpiece in accordance with the numerical control method,however, it is essential that the adjacent section curves should beconnected smoothly despite the absence of the data indicating the shapeof the surface between them. In other words, this means that machiningmust be performed by generating the curved surface lying between theadjacent section curves from such data as that indicative of the givensection curves, punching a numerical control (NC) tape so as to includethe data concerning the generated curved surface, and then machining theworkpiece in accordance with the instructions on the NC tape. To thisend, the NC tape ordinarily is prepared by using a computer, and eitherof two systems can be adopted to create the desired curved surfaces. Thefirst is a batch system in which processing is effected by dividing acurved surface into minute portions. In the second system atwo-dimensional curve made up of straight lines and arcs is modified foreach minute displacement applied to a third axis, i.e. the axis of thethird dimension. The first system, however, entails the processing oflarge quantities of data as well as highly complicated mathematicalprocessing, and requires a large-scale computer system. The secondsystem makes processing with a small scale computer possible, but thereis no three-dimensional tool offset capability and an excessivelimitation upon tool movement direction and machining shape, making itimpossible to create sophisticated three-dimensional bodies.

SUMMARY OF THE INVENTION

Accordingly, it is an object of the present invention to provide a novelmethod of creating curved surfaces that allows sophisticatedthree-dimensional bodies to be readily created while permittingprocessing to be performed by a small-scale computer.

Another object of the present invention is to provide a method ofcreating curved surfaces through which it is possible to create anentire body from data indicative of the shapes of sections drawn out ona blueprint, of a part of the three-dimensional body such as a metalmold.

A further object of the present invention is to provide a method ofcreating curved surfaces smoothly and accurately.

Yet another object of the present invention is to provide a method ofcreating curved surfaces through which machining can be performedaccurately on the basis of the created curved surfaces.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1, composed of (A), (B) and (C) is a diagram useful in describingthe creation of a curved surface according to the present invention;

FIG. 2, composed of (A), (B), (C), and (D) is a diagram useful indescribing the generation of intermediate sections according to thepresent invention;

FIG. 3, composed of (A), (B) and (C) is a diagram useful in describingthe generation of an intermediate section curve according to the presentinvention;

FIG. 4, composed of (A), (B) and (C) is a diagram useful in describing adata input method, FIG. 4A being a perspective view of a curved surface,and FIGS. 4B and 4C showing the section curves;

FIG. 5 is a diagram useful in describing a first method of generating anintermediate section;

FIG. 6 is a diagram useful in describing the coordinate system for thesections in FIG. 5;

FIG. 7 is a diagram useful in describing a second method of generatingan intermediate section;

FIG. 8 is a diagram useful in describing a third method of generating anintermediate section;

FIG. 9 is a diagram useful in describing a fourth method of generatingan intermediate section;

FIG. 10, composed of (A), (B), (C), (D), (E) and (F); is a diagramuseful in describing a first method of generating an intermediatesection curve;

FIG. 11, composed of (A), (B), (C), (D) and (E); is a diagram useful indescribing a second method of generating an intermediate section curve;

FIG. 12, composed of (A), (B), (C), (D) and (E) is a diagram useful indescribing a third method of generating an intermediate section curve;

FIG. 13, FIGS. 14A and 14B illustrate curved surfaces created inaccordance with the present invention;

FIG. 15 is a block diagram of a curved surface generating apparatus forrealizing the method of the present invention;

FIG. 16 is a block diagram of a device for generating intermediatesections according to the present invention;

FIG. 17 is a block diagram of a device for generating intermediatesection curves according to the present invention.

FIG. 18 is a block diagram of a conventional general purpose computerused for controlling a machine tool according to the present invention;and

FIG. 19 is a flow chart for the sections of processing performed by thepresent invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

A curved surface when drawn out on the plane of a blueprint of athree-dimensional metal mold or the like is generally represented by aplurality of section curves, and there is no data given for the shapesof the areas lying between a certain section curve and the adjacentsection curve. In effecting numerically controlled machining it isrequired that a metal mold be machined so as to smoothly connect thesetwo section curves despite the absence of the data indicating the shapeof the surface lying between the two section curves. In other words,machining must be performed by creating or generating the surface lyingbetween the two section curves from such data as that indicative of thegiven section curves, punching the data relating to the generatedsurface into an NC tape, and then machining the metal mold as specifiedby the instructions on the NC tape.

Reference will first be had to FIG. 1 to describe the creation of theabove-mentioned surface in accordance with the present invention. Thesteps are:

(1) generating a plurality of intermediate sections 14 and 15 betweentwo given sections 11 and 12, as shown in FIG. 1A;

(2) generating intermediate section curves 14a, 15a . . . , lying inrespective ones of the intermediate sections generated in step (1), fromsection curves (referred to as "given" section curves) 11a and 12a thatlie in the given sections 11 and 12, respectively, as shown in FIG. 1B;and

(3) generating the curved surface in the form of a continuous series ofintermediate section curves upon finding the intermediate section curves14a, 15a, . . . in a number of intermediate sections, as shown in FIG.1C.

The method of the present invention is therefore based upon generatingboth intermediate sections and intermediate section curves which lie inthe intermediate sections.

There are four methods of generating intermediate sections in accordancewith the given sections and given section curves drawn out on the planeof a blueprint. The four methods, illustrated in FIG. 2, are as follows:

(1) generating intermediate sections 13, 14, 15 and 16 in such a mannerthat they lie in parallel with given sections 11, 12 (referred to as afirst method of generating intermediate sections as shown in FIG. 2A);

(2) generating intermediate sections 13, 14, 15 and 16 in such a mannerthat they lie perpendicular to a given reference section 21 and, at thesame time, to a reference curve 21a which is a given section curve thatspecifies the external form of the curved body (referred to as a secondmethod of generating intermediate sections as shown in FIG. 2B);

(3) generating intermediate sections 13, 14 . . . in such a manner thatthey lie perpendicular to the given reference section 21 and at the sametime pass through a straight line 31 which is perpendicular to thereference section 21, the straight line 31 serving as an axis from whichthe intermediate sections 13, 14 . . . radiate (referred to as a thirdmethod of generating intermediate sections as shown in FIG. 2C); and

(4) generating intermediate sections 13, 14, 15 . . . in such a mannerthat they lie perpendicular to one of two given reference sections 21and 22 (referred to as a fourth method of generating intermediatesections as shown in FIG. 2D).

Furthermore, there are three methods of generating intermediate sectioncurves in accordance with the given section curves similarly drawn outon the plane of the blueprint. The three methods, illustrated in FIG. 3,are as follows:

(1) generating an intermediate section curve 13a in the intermediatesection 13 from the two given section curves 11a and 12a and onereference curve 21a (referred to as a first method of generatingintermediate section curves as shown in FIG. 3A);

(2) generating the intermediate section curve 13a in the intermediatesection 13 from one section curve 11a and two reference curves 21a and22a (referred to as a second method of generating intermediate sectioncurves as shown in FIG. 3B); and

(3) generating the intermediate section curve 13a in the intermediatesection 13 from the two given section curves 11a and 12a and tworeference curves 21a and 22a (referred to as a third method ofgenerating intermediate section curves as shown in FIG. 3C).

The foregoing is a general description of the inventive method ofcreating curved surfaces. With this as a background, the presentinvention will now be described in greater detail.

The present invention generates surfaces from a plurality of givensection curves, these being classified into "reference curves" and"section curves" depending on differences in their manner of use.Specifically, a "section curve" is a given curve upon which thegeneration of an "intermediate section curve" is based when creating acurved surface. A "reference curve" is a given curve utilized to decidethe special position of an intermediate section and to determine theshape of an intermediate section curve. The plane which contains thereference curve is referred to as a "reference section", and that whichcontains the section curve as a "section".

When creating a curved surface, a processor, which will be describedlater, must be supplied with input data, specifically section datarelating to the sections and reference sections represented on the planeof a blueprint, and section curve data relating to the section curvesand reference curves which are also drawn out on the blueprint. A datainput method in accordance with the present invention will now bedescribed assuming that the curved surface to be created is the curvedsurface SS₁ shown in FIG. 4A.

[I] Data input processing

(1) Inputting of section curves CV₁, CV₂, CV₃ and CV₄

(a) A section curve CV₁ contained in a section A is subjected to atransformation to place it in an H-V plane, as shown in FIG. 4B. Herethe section curve CV₁ shall be comprised of a series of three connectedarcs C₁, C₂ and C₃, and the start point and end point of each arc shallbe represented by (P₀, P₁), (P₁, P₂), (P₂, P₃), respectively.

(b) After CV₁ has been adopted as the term for the section curve insection A, the section curve CV₁ is defined in the following manner andthen applied to the processor as an input:

    ______________________________________                                                    CV.sub.1 = *                                                                 P.sub.0 (. . .)                                                               C.sub.1 (. . .), P.sub.1 (. . .)                                              C.sub.2 (. . .), P.sub.2 (. . .)                                              C.sub.3 (. . .), P.sub.3 (. . .).                                  ______________________________________                                    

The center and radius of an arc C_(i)(i=1,2,3) is specified within theparentheses following C_(i)(i=1,2,3), and the coordinates of the pointP_(i) are specified in the parentheses following P_(i).

(c) Finally, the terms CV₂, CV₃ and CV₄ are similarly assigned to thesection curves contained in the sections B, C, and D, respectively, andthe section curves are defined as described above and then entered intothe processor.

It should be noted that while three connected arcs were taken asconstructing the section curve CV₁ contained in the section A, a sectioncurve CV_(i) generally is not limited to a series of arcs alone but isdefined as a curve composed of line segments, arcs and a given series ofpoints and the like that are smoothly connected. FIG. 4C is an exampleof such a section curve. Here a line segment S₁ extending from a pointP₁₀ to a point P₁₁, an arc C₁ extending from the point P₁₁ to a pointP₁₂, and a curved segment T₁ extending from the point P₁₂ to the pointP₁₈ through a series of points P₁₃ through P₁₇, are smoothly connectedto form a curve.

After the section curves CV_(i)(i=1, 2, 3, 4) in respective sectionshave been defined by observing the above-mentioned steps (a) through (c)and then applied to the processor as inputs, the curved surface SS₁ isdefined as shown below using the aforesaid section curves, and is thenapplied to the processor.

(2) Inputting of surface SS₁

The inputting of the curved surface SS₁ is performed after the curvedsurface has been defined in the following manner:

SS₁ =201, BC₁ (CV₁, YZ, 100),, BC₂ (CV₂, YZ), DC₁ (CV₃, XZ, 80),, DC₂(CV₄, XZ).

Here the number 201 is a type number that represents the type of curvedsurface that is to be generated; it allows curved surfaces of varioustypes to be specified depending upon the particular objective. BC₁(CV₁,YZ, 100) represents a reference curve and shows that the curve CV₁lies in a section which is parallel to the YZ plane, that the sectioncuts the X-axis at a point x=100, and that the curve CV₁ has alreadybeen defined. Similarly, BC₂ (CV₂, YZ) means that the curve CV₂ is areference curve on the YZ plane. DC₁ (CV₃, XZ, 80) and DC₂ (CV₄, XZ)denote section curves and indicate that the curves CV₃, CV₄ each lie inparallel with the XZ plane, the curve CV₃ cutting the Y-axis at a pointY=80, the curve CV₄ lying on the XZ plane.

In the case described above, BC₁, BC₂, DC₁ and DC₂ all lie on thecoordinate planes or on planes which lie in parallel with the coordinateplanes. However, spatially oblique planes can be specified by changingthe indications within the parentheses that follow BC₁, BC₂, DC₁ andDC₂.

Thus, the section data and section curve data necessary for generatingthe curved surface SS₁ is input through the steps (1) and (2) describedabove.

The processor, once provided with the input data, begins processing tocreate the curved surface. This is done by performing the process forintermediate section generation, for intermediate section curvegeneration and finally, for generation or creation of the desired curvedsurface.

[II] Processing for intermediate section generation

An intermediate section is generated by the first, second, third orfourth method of intermediate section generation, described above, inaccordance with given section data and given section curve data obtainedfrom a blueprint (which data has already been fed into the processor bythe data input processing step [I] set forth above). Further, theintermediate section is specified by "intermediate section information".The latter contains "section information" which indicates what thenumber of the obtained intermediate section in the desired surface is,and "section position information", such as a transformation matrix,which indicates how to transform a specific coordinate system to obtainthe derived intermediate section.

The methods of generating an intermediate section will be describednext.

(1) First method (Method I) of generating an intermediate section

FIG. 5 illustrates the first method of generating an intermediatesection. Given sections which are mutually parallel are shown atnumerals 11 and 12, an intermediate section at numeral 13, a referencesection at 21, given section curves at 11a, 12a, and a reference curveat 21a. The intermediate section 13 is generated through the followingsequence:

Step (1) A point P₁ ", which divides the length of the reference curve21a into a ratio of m:n, is computed. The point lies at the intersectionof the intermediate section 13 and the reference curve 21a. The ratiom:n is the "section information" mentioned above.

The followings steps (1-1) through (1-4) are preformed to find the pointwhich divides a given reference curve 21a into a ratio of m:n from oneend point P₁ :

Step (1-1): The length of each element constituting the given referencecurve 21a is found (where the term "element" is taken to mean a linesegment or an arc), and the lengths obtained are added together to findthe total length D of the curve.

Step (1-2): An equation ##EQU1## is set up.

Step (1-3): This step is the extraction of an element containing a pointwhere distance from the one end point P₁ is D'. If D₁ is taken as thelength of the initial element, D₂ as the length of the next element.,D_(i) as the length of the i-th element and so on, the extraction of theelement is carried out by finding the k that satisfies the followinginequality: ##EQU2##

Step (1-4): This step is to find the point on the k-th element whosedistance from starting point of the k-th element is D", as shown in thefollowing equation, ##EQU3## The obtained point is that which dividesthe given reference curve 21a into a ratio of m:n from one end thereof.In Step (1-3), when k=1, it shall be assumed that ##EQU4##

Step (2)

This step is the computation of a transformation equation which isutilized for obtaining the intermediate section 13 passing through thepoint P₁ " obtained in Step (1) and lying in parallel with the givensection 11. The transformation equation is found from the section dataof the given section 11, the coordinates of the dividing point P₁ ", andfrom the information indicating which of the first through fourthmethods of intermdiate section generation was used to generate theintermediate section 13.

By imagining a coordinate system on the intermediate section 13 asformed by the following procedure, these can be computed, atransformation equation which transforms the plane of a specificcoordinate system (such as the XY plane) into the imagined coordinatesystem on the intermediate section in space. This transformationequation (transformation matrix) serves as the section positioninformation. The procedure for obtaining the imagined coordinate systemis as follows:

Step (2-1): It is assumed that the reference section 21 and intermediatesection 13 intersect in a line HL, this line will be divided into twosegments by the point of intersection P₁ " of the reference curve 21aand the intermediate section 13, as shown in FIG. 5. Of the two linesegments that result from this division, one is chosen in accordancewith a certain fixed rule. For example, when viewing from the directionof progress of the reference curve 21a, the line segment of the righthand side is chosen [segment ○a in FIG. 6A].

Step (2-2): In a similar manner, a straight line VL, which isperpendicular to the reference section 21 and which passes through thepoint of intersection P₁ " of the reference curve 21a and theintermediate section 13, is divided into two segments by the referencesection 21. Of the two resulting segments, one is chosen in accordancewith a certain fixed rule. For example, the reference section 21 isalways viewed from the +Z direction, and the segment on the far side ischosen [segment ○b in FIG. 6A].

Step (2-3): In this step a planar coordinate system, wherein the pointof intersection P₁ " of the reference curve 21a and the intermediatesection 13 is the origin and wherein the directions ○a and ○b in FIG. 6are the directions of the two axes (the coordinate system is shown inFIG. 6B), is taken as the planar coordinate system of the intermediatesection 13.

(2) Second method (Method II) of generating an intermediate section

FIG. 7 is an illustrative view showing the second method of generatingan intermediate section. Reference numeral 21 denotes the referencesection, 21a a reference curve lying in the reference section 21 andspecifying the external form of a body, and 13 the intermediate section.The intermediate section 13 is generated through the following sequence:

Step (1)

The point P₁ " is found through the same procedure outlined in Step (1)of the first method (Method I) of generating an intermediate section.The point P₁ " is that point which divides the length of the referencecurve 21a, between the end points P₁, P₁ ' thereof, into a ratio of m:n.The ratio m:n serves as the section information.

Step (2)

A normal NL is formed, which is perpendicular to the reference curve 21aat the dividing point P₁ " found in Step (1). Next there is computed atransformation equation for effecting a transformation from a specificcoordinate system to obtain the intermediate section 13 which isperpendicular to the reference section 21 and which contains the normalNL. This transformation equation is computed through the same procedureas that described in Step (2) of the first method (Method I) ofgenerating an intermediate section.

(3) Third method (Method III) of generating an intermediate section

FIG. 8 is an illustrative view showing the third method of generating anintermediate section. Reference numeral 21 denotes the referencesection, 21a a reference curve lying in the reference section 21 andspecifying the external form of a body, 31 an axis perpendicular to thereference section 21, and 13 the intermediate section.

Step (1)

The point P₁ " is found through the same procedure outlined in Step (1)of the first method (Method I) of generating an intermediate section.The point P₁ " is that point which divides the length of the referencecurve 21a, between the end points P₁ and P₁ ' thereof, into a ratio ofm:n. The ratio m:n serves as the section information.

Step (2)

There is computed a transformation equation for effecting atransformation from a specific coordinate system to obtain theintermediate section 13 that passes through the axis 31 and the dividingpoint P₁ " found in Step (1), and that at the same time is perpendicularto the reference section 21. This transformation equation is computedthrough the same procedure as that described in Step (2) of the firstmethod (Method I) of generating an intermediate section.

(4) Fourth method (Method IV) of generating an intermediate section

FIG. 9 is an illustrative view showing the fourth method of generatingan intermediate section. Reference numerals 21 and 22 denote first andsecond reference sections, respectively, reference numerals 21a and 22adenote first and second reference curves, respectively, which lie in thereference sections 21 and 22 and which specify the external form of abody, and reference numeral 13 denotes the intermediate section.

Step (1)

Points P₁ " and P₂ " are found through the same procedure outlined inStep (1) of the first method (Method I) of generating an intermediatesection. The points P₁ " and P₂ " are those that divide lengths of therespective reference curves 21a and 22a into a ratio of m:n. The ratiom:n serves as the section information.

Step (2)

A transformation equation is computed. This is for a transformation froma specific coordinate system in order to obtain the intermediate section13 which contains the points P₁ " and P₂ " and which contains the pointof intersection P₃ " of the first reference section 21 and aperpendicular dropped from the dividing point P₂ " to the firstreference section 21. The equation is computed through the sameprocedure as that described in Step (2) of the first method (Method I)generating an intermediate section.

After the intermediate section 13 has been generated by Method I, II,III or IV described above, processing is executed for the generation ofan intermediate section curve lying in the intermediate section. [III]Processing for intermediate section curve generation

An intermediate section curve is generated by one of the following threemethods of intermediate section curve generation in accordance withinformation indicating which one of Methods (I) through (IV) in theprocessing description [II] has been used to generate the intermediatesection, and in accordance also with the given section curve data of theparticular section on the blueprint. The procedure for generating anintermediate section curve comprises the following four steps:

(a) obtaining the section information which is the dividing ratio m:n ofthe desired intermediate section;

(b) transforming each one of given section curves and or intervalsformed by two reference curves, so that they lie on a same plane;

(c) generating an intermediate section curve that lies on said plane;and

(d) transforming the intermediate section curve on said plane into onewhich lies in the desired intermediate section.

(1) First method (Method I) of generating an intermediate section curve

The first method of intermediate section curve generation is applicableto a case in which the data relating to two section curves and onereference curve is given. As long as this data is known, theintermediate section curve lying in the intermediate section can begenerated regardless of which one of Methods I through III is employedto generate the intermediate section.

FIG. 10 is an illustrative view showing the first method (Method I) ofgenerating an intermediate section curve. Reference numerals 11 and 12denote two given sections of a body, and 11a and 12a denote givensection curves for a case in which the body is cut by the given sections11 and 12. Reference numeral 21 denotes a reference section containingpoints P₁ and P₁ ' on the respective section curves 11 and 12, 21a areference curve lying in the reference section 21 and specifying theexternal form of the body, and 13 an intermediate section generated byMethod II of intermediate section generation so as to contain a dividingpoint P₁ " which divides the length of the reference curve 21a into aratio of m:n. It should be noted, however, that the intermediate section13 could just as well have been generated by either Method I or MethodIII for intermediate section generation.

The procedure for generating the intermediate section curve by the firstmethod (Method I) of generation will now be described with reference toFIG. 10.

Step (1)

Section information which is the dividing ratio m:n concerning thedesired intermediate section 13 is obtained.

Step (2)

The given section curves 11a and 12a are transformed onto the same plane(FIG. 10B). The given section curves 11a, 12a can be considered to becurves which lie on the same plane if they are manipulated by thefollowing steps (2-1) through (2-3):

Step (2-1): The points of intersection P₁ and P₁ ' of the referencecurve 21a and the two given sections 11 and 12 are taken as the samepoint.

Step (2-2): When the lines of intersection HL and HL' of the referencesection 21 and the two given sections 11 and 12 are considered, it isseen that the both lines of intersection HL and HL' are divided by thepoints of intersection P₁ and P₁ ', respectively, of these partitionedsegments, those that have the same direction (A and A') with respect tothe reference curve 21a are superposed.

Step (2-3): When straight lines VL and VL', which pass through thepoints of intersection P₁ and P₁ ' of the reference curve 21a and thetwo given sections 11 and 12, and which at the same time areperpendicular to the reference curve 21a, and considered, it is seenthat these lines of intersection VL and VL' are partitioned by thepoints of intersection P₁ and P₁ ', respectively. Of these partitionedsegments, those that have the same direction (B and B') with respect tothe reference curve 21a are superposed. Thus, the given section curves,now denoted by 11a' and 12a', are transformed into curves on the sameplane. See FIG. 10B.

Step (3)

By using the two given section curves 11a' and 12a' lying in theAB-plane obtained from Step 2 above, an intermediate section curve 13a'is generated in said plane.

The intermediate section curve 13a' is generated through the followingprocedure:

Step (3-1): Using the steps (1-1) through (1-4) described above inconnection with the Method I for intermediate section generation, pointsQ₁ and Q₂, which divide the lengths of the respective given sectioncurves 11a' and 12a' each into a ratio of a:b, are computed (FIG. 10C).

Step (3-2): A dividing point Ri is computed, which point divides astraight line connecting the dividing points Q₁ and Q₂ into a ratio ofm:n of Step (1) (FIG. 10D).

If the coordinates of the dividing points Q₁ and Q₂ are denoted by (x₁,y₁) and (x₂, y₂), respectively, then the coordinates of a dividing pointRi are calculated from: ##EQU5##

Step (3-3): The intermediate section curve 13a' is generated by a seriesof points Ri (i=1, 2, . . . ) obtained by changing the value of thedividing ratio a/b of step (3-1) gradually from 0 to 1 (FIG. 10E). Asmoother intermediate section curve 13a' can be obtained by making thesuccessive changes in the value of a/b very small.

Step (4)

The intermediate section curve 13a' on the AB-plane, which section curve13a' was obtained from Step (3) above, is transformed into a curve lyingin the defined spacial intermediate section 13 (FIG. 10A). The equationfor the transformation which transforms the AB-plane obtained from Step(2) into the intermediate section 13 can be expressed by a combinationof the parallel translation and rotation in space and generally isexpressed by the matrix M shown below. Accordingly, by setting up thetransformation matrix M for the points Ri(i=1, 2, . . . ) obtained inStep (3), the points Ri can be transformed into points in the definedspecial intermediate section 13. The curve connecting the series ofpoints in the defined intermediate section 13 obtained by the matrixtransformation is the intermediate section curve 13a (FIG. 10F).

The matrix M expressing the transformation equation has the followingform: ##EQU6## where a₂₂, a₂₃, a₃₂, a₃₃ indicate rotation about theX-axis, a₁₁, a₁₃, a₃₁, a₃₃ indicate rotation about the Y-axis, a₁₁, a₁₂,a₂₁, a₂₂ indicate rotation about the Z-axis, a₄₁ indicates translationparallel to the X-axis, a₄₂ translation parallel to the Y-axis, and a₄₃translation parallel to the Z-axis. Accordingly, a point (x, y, z, 1)prior to transformation is transformed into a point (X, Y, Z, 1) in thedefined intermediate section by multiplying it together with thetransformation equation M, that is,

    (X, Y, Z, 1)=(x, y, z, 1)·M.

The transformation matrix M is obtained by finding the matrix thattransforms the origin P₁ on the predetermined plane into thecorresponding point P₁ " in the intermediate section.

(2) Second method (Method II) of generating an intermediate sectioncurve

This method is applicable to a case where the available data is the datarelating to one given section curve and two reference section curves.The intermediate section curve lying in the intermediate section can begenerated regardless of which one of Methods I through IV forintermediate section generation is employed.

FIG. 11 is an illustrative view showing the second method of generatingan intermediate section curve. Reference numeral 11 denotes a givensection of a body, 11a a given section curve for a case in which thebody is cut by the given section 11, and 21 and 22 denote first andsecond reference sections containing the points P₁ and P₂, respectively,that lie on the given section curve 11a. Reference numerals 21a and 22adenote reference curves which lie in the respective first and secondreference sections 21 and 22 and which specify the external form of thebody. Reference numeral 13 denotes an intermediate section generated byMethod IV for intermediate section generation which contains points P₁ "and P₂ " that divide the respective first and second reference curves21a and 22a internally into a ratio of m:n, and which contains the pointof intersection P₃ " of the first reference section 21 and aperpendicular from the dividing point P₂ " to the first referencesection 21.

The procedure for generating the intermediate section curve by thesecond method (Method II) of generation will now be described withreference to FIG. 11.

Step (1)

Section information which is the dividing ratio m/n concerning thedesired intermediate section 13 is obtained.

Step (2)

The given section curve 11a and the points of intersection (the dividingpoints for establishing the ratio m:n) P₁ " and P₂ " of the intermediatesection 13 and the first and second reference curves 21a and 22a aretransformed into a curve 11a' and points P₁ " and P₂ " on the sameAB-plane (FIG. 11B). This transformation is performed through the sameprocedure consisting of the Steps (2-1) through (2-3) described above inconnection with the Method I for intermediate section curve generation.

Step (3)

The given section curve 11a' and the points of intersection P₁ " and P₂" on the AB-plane, as obtained in Step (2), are used to generate anintermediate section curve lying on said plane.

The intermediate section curve is generated through the followingprocedure:

Step (3-1): A computation is performed to obtain the ratio l₁ /l₂ of thelength of the line segment connecting the starting point P₁ with endpoint P₂ of the given section curve 11a' resulting from thetransformation into the predetermined plane, to the length of the linesegment connecting the points of intersection P₁ " and P₂ ' which havealso been transformed into points in the predetermined plane. Inaddition, an angle θ defined by P₂ P₁ P₂ " is computed. The angle θ isthe angle of rotation through which the line segment P₁ P₂ is rotated inthe counterclockwise direction to bring it into coincidence with theline segment P₁ " P₂ ". The counterclockwise direction is taken as thepositive direction (FIG. 11C).

Step (3-2): The curve dividing point Si that divides the given sectioncurve 11a' into a ratio of a:b is computed according to the method ofSteps (1-1) through (1-4) as described in connection with the Method Ifor intermediate section generation (FIG. 11C).

Step (3-3): A computation is performed to find a curve point Si" whichresults when an external curve dividing point Si', for externallydividing the line segment P₁ Si into the ratio of l₁ :l₂ is rotated bythe angle θ (FIG. 11C).

Letting (x_(i), y_(i)) represent the coordinates of the curve dividingpoint Si that divides the given section curve 11a' into a ratio of a:b,letting (x_(o), y_(o)) represent the coordinates of the point P₁, andletting (X, Y) represent the coordinates of the point Si", thecoordinates of the point Si" are found from: ##EQU7##

Step (3-4): The intermediate section curve 13a' is generated (FIG. 11D)by a series of points Si" (i=1, 2, 3 . . . ) by changing the value ofthe dividing ratio a/b of step (3-2) gradually from 0 to 1.

Step (4)

The intermediate section curve 13a' on the AB-plane as obtained fromStep (3) above is transformed into a intermediate curve 13a lying in thedefined spacial intermediate section 13 (FIG. 11A). The transformationmethod is the same as that described in Step (4) in connection with thefirst method I for intermediate section curve generation.

In the foregoing the intermediate section 13 was generated by Method IVfor intermediate section generation, and the intermediate section curvewas generated in said intermediate section. This Method II forintermediate section curve generation can, however, be applied to caseswhere the intermediate section is generated by any one of the Methods Ithrough III for intermediate section generation as well.

FIG. 11E is an illustrative view for describing the generation of anintermediate section curve in an intermediate section generated by theMethod II of intermediate section generation. Portions corresponding tothose shown in FIG. 11A are denoted by like reference characters andwill not be described in detail again. In FIG. 11E, the intermediatesection 13 is generated by the second Method II of intermediate sectiongeneration so as to contain the point P₁ " that divides the firstreference curve 21a internally into a ratio of m:n. The procedure forgenerating the intermediate section curve in this case will now bedescribed with reference to FIG. 11E.

Step (1)

Section information which is the dividing ratio m/n concerning thedesired intermediate section 13 is found.

Step (2)

The point of intersection P₂ " of the intermediate section 13 and thesecond reference curve 22a is found.

Step (3)

The intermediate section curve 13a is found by carrying out the steps(2) through (4) of the Method II for intermediate section curvegeneration described above.

(3) Third method of generating an intermediate section curve

This method is applicable to a case where the available data is the datarelating to two given section curves and two reference curves. Theintermediate section curve lying in the intermediate section can begenerated regardless of which of the Methods I through IV forintermediate section generation is employed.

FIG. 12 is an illustrative view showing the third method (Method III) ofgenerating an intermediate section curve. Reference numerals 11 and 12denote given sections of a body, and 11a and 12a denote given sectioncurves for a case in which the body is cut by the given sections 11 and12. Numeral 21 represents a first reference section which contains firstpoints P₁ and P₁ ' lying on the given section curves 11a and 12a,respectively, 22 a second reference section which contains second pointsP₂, P₂ ' lying on the given section curves 11a and 12a, respectively,and 21a and 22a denote first and second reference curves, respectively,which lie in the respective first and second reference sections 21 and22 and which specify the external form of the body. Reference numeral 13denotes an intermediate section generated by the Method IV forintermediate section generation which contains points P₁ " and P₂ " thatdivide the respective first and second reference curves 21a and 22ainternally into a ratio of m:n, and which contains the point ofintersection P₃ " of the first reference plane 21 and a perpendicularfrom the dividing point P₂ " to the first reference section 21.

The procedure for generating the intermediate section curve by theMethod III for section curve generation will now be described withreference to FIG. 12. Here the Method III is a combination of theMethods I and II for intermediate section curve generation.

Step (1)

Section information which is the dividing ratio m/n concerning thedesired intermediate section 13 is found.

Step (2)

The given section curves 11a, 12a and the points of intersection P₁ "and P₂ " of the intermediate section 13 with the first and secondreference curves 21a and 22a are transformed into curves and points onthe same plane (AB-plane) (FIG. 12B). This transformation is performedthrough the same procedure consisting of the Steps (2-1) through (2-3)described above in connection with the Method I for intermediate sectioncurve generation.

Step (3)

The given section curves 11a' and 12a' on the AB-plane as obtained inStep (2) are used to generate a intermediate section curve 13b lying onsaid AB-plane. This intermediate section curve 13b is generatedaccording to the same procedure consisting of the Steps (3-1) through(3-3) described in connection with the Method I for intermediate sectioncurve generation (FIG. 12C).

Step (4)

The intermediate section curve 13b and the points of intersection P₁ "and P₂ " on the AB-plane as obtained in Step 3 above are used togenerate an intermediate section curve 13a' on said AB-plane. Thisintermediate section curve 13a' is generated according to the sameprocedure consisting of the steps (3-1) through (3-4) described inconnection with the Method II for intermediate section curve generation(FIG. D).

Step (5)

The intermediate section curve 13a' on the AB-plane obtained from Step(4) above is transformed into a curve lying in the defined spacialintermediate section 13 (FIG. 12A). The transformation method is thesame as that described in Step (4) in connection with the Method I forintermediate section curve generation.

In the foregoing the intermediate section was generated by the Method IVfor intermediate section generation, and the intermediate section curvewas generated in said intermediate section. The Method III forintermediate section curve generation can, however, be applied to caseswhere the intermediate section is generated by any of the Methods Ithrough III for intermediate section generation as well.

FIG. 12E is an illustrative view for describing the generation of anintermediate section curve in an intermediate section generated by theMethod II for intermediate section generation. Portions corresponding tothose shown in FIG. 12A are denoted by like reference characters andwill not be described in detail again. In FIG. 12E, the intermediatesection 13 is so generated by the Method II for intermediate sectiongeneration as to contain the point P₁ " that divides the first referencecurve 21a internally into a ratio of m:n. The procedure for generatingthe intermediate section curve in this case will now be described withreference to FIG. 12.

Step (1)

Section information which is the dividing ratio m/n concerning thedesired intermediate section 13 is found.

Step (2)

The point of intersection P₂ " of the intermediate section 13 and thesecond reference curve 22a is found.

Step (3)

The intermediate section curve 13a is generated by carrying out thesteps (2) through (4) of the Method III for intermediate section curvegeneration described above.

[IV] Processing for curved surface generation

A number of the given section curves 13a are obtained when the foregoingprocessing for intermediate section generation and processing forintermediate section curve generation are repeated while successivelychanging the dividing ratio m:n from 0:1 to 1:0. A smooth curved surfacecan then be generated by connecting the obtained intermediate sectioncurves. A smoother curved surface can be obtained by making thesuccessive changes in the ratio m:n very small. FIG. 13 shows an exampleof a surface generated by the method of the present invention. Even aplurality of continuous curved surfaces can be generated. In such asituation, applying the invention method of curved surface generation toindividual ones of the curved surfaces makes it possible to generate asmooth, continuous curved surface of the type shown in FIGS. 14A and14B. The latter illustrates the sculptured surface of a female torsodrawn by applying the present invention.

[V] Machining the Curved Surface

When machining data relating to the curved surface of athree-dimensional body is obtained as described above, then the datarelating to the curved surface is punched into an NC tape as describedon page 6, lines 21 to 27. Thus, punched NC tape is used as instructionsfor machining the metal mold.

FIG. 15 shows a conceptual block diagram of the flow within an apparatusfor generating curved surfaces in order to practice the presentinvention. The apparatus includes a data input unit 301 for inputtingdata such as that representing the given sections and section curvesdrawn on the plane of a blueprint, and a memory 302 for storing thegiven section and section curve data as well as data such as thatrepresenting intermediate sections and intermediate section curvesgenerated by a conceptual intermediate section generating device 303 andby a conceptual intermediate section curve generating device 304. Theintermediate section generating device 303, using the section data andsection curve data stored in the memory 302 as well as informationindicative of which one of the first through fourth methods ofintermediate section generation is to be used to generate anintermediate section, generates the particular intermediate section inaccordance with the steps constituting the selected method ofintermediate section generation and stores the intermediate section datarelating to this intermediate section in the memory 302. Theintermediate section curve generating device 304, using the sectioncurve data, section data, intermediate section data as well asinformation indicative of which one of the first through third methodsof intermediate section curve generation is to be used to generate anintermediate section curve, generates the intermediate section curve inaccordance with the steps constituting the selected method ofintermediate section curve generation and stores the intermediatesection curve data relating to this intermediate section curve in thememory 302. A tape preparation control unit 305 reads from the memory302 such data as the stored intermediate section data and stored datarelating to the intermediate section curves lying on the intermediatesections, and delivers this data to a tape puncher 307 and printer 309upon converting it into an NC tape format for machining a metal mold orthe like. A control circuit 306 controls the memory 302, intermediatesection generating device 303, intermediate section curve generatingdevice 304 and tape preparation control unit 305. Numeral 308 denotes apaper tape punched by the puncher 307.

The above apparatus for generating curved surfaces sequentially producesintermediate section curves in accordance with the processing steps forintermediate section generation and for intermediate section curvegeneration, and converts the intermediate section curve data into a tapeformat before delivering it to the tape puncher 307 and printer 309.

A conceptual block diagram of the flow for the conceptual intermediatesection generating device 303 is shown in FIG. 16. The device includes aconceptual dividing point computing unit 303a for computing thecoordinates of the dividing point P₁ " (such as shown in FIG. 10A) uponreceiving data BCD specifying the reference curve 21a, a dividing numberM and dividing ratio data DR, and a dividing ratio storage register 303bfor storing new dividing ratio data DR each time the generation of oneintermediate section curve has been completed. More specifically, thedividing point computing unit 303a derives the dividing ratio m:n byperforming the computations i+1→m and M-m →n and delivers this as thedividing ratio data to the dividing ratio storage register 303b, andfinds the coordinates of the dividing point for establishing this ratioin accordance with the Steps (1-1) through (1-4) described above inconnection with Method I of generating an intermediate section. Theinitial value of i in the above computation is 1, and the processing forintermediate section generation is executed until such time thati=(M-1). The device includes also an intermediate section generatingunit 303c which generates intermediate sections by employing referencesection data BSD, reference curve data BCD and section generation dataSCD indicative of which one of the first through fourth methods ofintermediate section generation is to be employed for generating thesection. In other words, assuming that an intermediate section is to begenerated by the Method II, the conceptual generating unit 303c willcompute and deliver intermediate section data MSD relating to a sectionwhich contains the dividing point P₁ " and which is at the same timeperpendicular to both a reference section and a reference curve.

A conceptual block diagram of the flow for the conceptual intermediatesection curve generating device 304 is illustrated in FIG. 17. Thedevice includes a conceptual section curve transformation processingunit 304a whose inputs are section curve data SCC and section data SSD.Assuming that the input data represents the two section curves and onereference section curve, the processing unit 304a executes processing todevelop the section curves on the same plane and to transform the givensection curve data into a coordinate system on said plane in accordancewith the Step (2-1) through (2-3) of the Method I of intermediatesection curve generation. The intermediate section curve generatingdevice 304 includes also conceptual intermediate section curve computingunit 304b and conceptual intermediate section curve transformationprocessing unit 304c. The computing unit 304b generates intermediatesection curves based on (a) the dividing ratio (m/n) stored in thedividing ratio storage register 303b of FIG. 16, and (b) control data CDindicative of which one of the first through third methods ofintermediate section curve generation is to be used to generate theparticular intermediate section curve. Assuming for example that anintermediate section curve is to be generated by the Method I, thecomputing unit 304b executes processing in accordance with Step (3) ofthe Method I and generates the intermediate section curve 13a' (FIG.10E) in the form of a continuous number of points Ri(i=1, 2 . . . ). Thetransformation processing unit 304c in this case executes processing forthe matrix transformation in Step (4) of the Method I and develops theintermediate section curve 13A', delivered by the intermediate sectioncurve computing unit 304b, on the intermediate section data MSDgenerated by the intermediate section generating unit 303c shown in FIG.16. The output of the transformation processing unit 304c isintermediate section curve data MSC which is stored sequentially in amemory device that is not shown. A plurality of the intermediate sectioncurves are processed in the above manner and combined to create a body.

It should be noted that the curved surface generating apparatus of FIG.15 can be composed of a computer system, in which the conceptual controlcircuit 306 can be replaced by a control program and curved surfacegeneration program and the conceptual intermediate section generatingdevice 303, conceptual intermediate section curve generating device 304and tape preparation control unit 305 by a processor.

FIG. 18 shows the conventional components of a conventional comutersystem that can perform the method of the present invention, asconceptionally shown by FIGS. 15-17. FIG. 19 shows the generalprocessing steps of the present invention as previously described indetail herein.

Thus, in accordance with the present invention as described above,curved surfaces can be created from a small quantity of data andprocessing for creation of the curved surface machining data can beexecuted with ease. This permits curved surfaces to be created even bysmall-scale computers. The invention makes it possible to create evensophisticated bodies in the form of a continuous series of pluralsurfaces.

As many widely different embodiments of this invention may be madewithout departing from the spirit and scope thereof, it is to beunderstood that the invention is not limited to the specific embodimentsthereof except as defined in the appended claims.

What we claim is:
 1. A method of machining a curved surface of athree-dimensional body on a workpiece using a digital computer,comprising the steps of:(a) inputting, into said computer, data relatingto a section which specifies given sections of said body, data relatingto a section curve which lies on said section and specifies the externalform of said body, data relating to a reference section which contains apoint lying on said section curve in said section and which liesperpendicular to said section which specifies said given sections ofsaid body, and data relating to a reference curve which lies on saidreference section and which specifies the external form of said body;(b) creating data relating to an intermediate section which liesperpendicular to said reference section on the basis of said data whichare input to said computer; (c) creating intermediate section curve datarelating to an intermediate section curve on the basis of said dataobtained in said steps (a) and (b); (d) creating machining data relatingto said curved surface of said body by repeating the said steps (a) and(c) after transferring a point of intersection of said intermediatesection curve and said reference curve; (e) generating blocks of controldata based on said machining data; (f) producing a paper tape includingsaid control data; and (g) controlling a machine to produce said curvedsurface of said three-dimensional body on the workpiece on the basis ofsaid control data on the paper tape.
 2. A method according to claim 1,wherein step (e) comprises the steps of:(i) punching a numerical controltape in accordance with said created machining data relating to saidcurved surface; and (ii) machining said curved surface of saidthree-dimensional body in accordance with instructions from said NCtape.
 3. A method according to claim 1, wherein step (b) furthercomprises the step of generating said intermediate sections that lie inparallel with said given sections.
 4. A method according to claim 1,wherein step (b) further comprises the step of generating each of saidintermediate sections where each lies perpendicular to said referencesection and to the reference curve which lies in said reference section.5. A method according to claim 1, wherein step (b) further comprises thestep of generating each of said intermediate sections where each liesperpendicular to said reference section and passes through a straightline which is perpendicular to said reference section, said straightline serving as an axis from which said intermediate sections radiate.6. A method according to claim 1, wherein step (a) further comprises thestep of inputting first and second reference sections containingrespective first and second reference curves which specify the externalform of the body, as said given sections.
 7. A method according to claim1, wherein step (b) further comprises the step of generating each ofsaid intermediate sections where each lies perpendicular to one of saidreference sections.
 8. A method of machining a curved surface of athree-dimensional body on a workpiece using a digital computer,comprising the steps of:(a) inputting, into said computer, section datarelating to two parallel sections which specify given sections of saidcurved surface of said three-dimensional body, and section data relatingto a reference section which contains a reference curve specifying theexternal form of the body and containing first and second points lyingon respective given section curves in respective ones of said givensections; (b) creating data relating to a plurality of given sections ofthe body in accordance with said section data; (c) dividing saidreference curve into a plurality of segments at dividing points andproducing dividing point data; (d) computing an equation for atransformation from a specific coordinate system to obtain anintermediate section which contains one of the dividing points on thereference curve and which lies in parallel with said parallel sections;(e) computing data relating to an intermediate section curve in saidintermediate section on the basis of data relating to the section curvesin respective ones of said parallel sections and on the basis of thedividing point data relating to the position of the one of said dividingpoints contained in said intermediate section; (f) computingintermediate curve section data relating to said intermediate sectioncurve of said intermediate section containing the one of said dividingpoints; (g) creating machining data relating to said curved surface ofsaid three-dimensional body from intermediate curve section datarelating to a plurality of intermediate section curves; (h) generatingblocks of control data based on said machining data; (i) producing apaper tape including said control data; and (j) controlling a machine toproduce said curved surface of said three-dimensional body on theworkpiece on the basis of said control data on the paper tape.
 9. Amethod according to claim 8, wherein step (b) comprises the steps of:(i)dividing said reference curve internally into a ratio of m:n; and (ii)generating said dividing points by changing the ratio m:n.
 10. A methodaccording to claim 9, wherein step (d) comprises the steps of:(i)transforming each section curve into a corresponding curve in apredetermined planar corrdinate system to bring into coincidence thefirst and second points and to bring into coincidence lines ofintersection of the reference section and said two parallel sections;(ii) computing an intermediate point by dividing internally into a ratioof a:b each section curve transformed into said predetermined planarcoordinate system, the dividing at the ratio a:b designating third andfourth points, and dividing internally into the ratio of m:n a straightline connecting the third and fourth points at which said section curvesare internally divided into the ratio a:b, the straight line dividingdesignating the intermediate point; (iii) computing an untransformedintermediate section curve in said predetermined planar coordinatesystem by computing a number of intermediate points by varying the ratioa:b; and (iv) transforming said untransformed intermediate section curvein said predetermined planar coordinate system into the correspondingintermediate section curve in said intermediate section.
 11. A method ofmachining a curved surface of a three-dimensional body on a workpieceusing a digital computer, comprising the steps of:(a) inputting, intosaid computer, section data relating to at least two sections of thebody, and data relating to a reference section which contains areference curve specifying the exteral form of the body and containingfirst and second points lying on respective section curves in respectiveones of said two sections; (b) specifying a plurality of given sectionsof the body in accordance with said section data; (c) dividing saidreference curve into a plurality of segments at a dividing point andproducing dividing point information; (d) computing an equation for atransformation from a specific coordinate system to obtain anintermediate section which contains one of the dividing points on thereference curve and which lies perpendicular to said reference sectionand to said reference curve; (e) computing data relating to anintermediate section curve in said intermediate section on the basis ofsaid dividing point information relating to the position of the one ofsaid dividing points contained in said intermediate section and on thebasis of data relating to the section curves in respective ones of saidtwo sections; (f) computing intermediate curve section data relating tosaid intermediate section curve of the intermediate section containingsaid dividing point; (g) creating machining data relating to said curvedsurface of said three-dimensional body on the basis of said intermediatecurve section data relating to a plurality of intermediate sectioncurves; (h) generating blocks of control data based on said machiningdata; (i) producing a paper tape including said control data; and (j)controlling a machine to produce said curved surface of said threedimensional body on the workpiece on the basis of the control data onthe paper tape.
 12. A method according to claim 11, wherein step (d)comprises the steps of:(i) dividing said reference curve internally intoa ratio m:n; and (ii) generating said dividing points by changing theratio m:n.
 13. A method according to claim 12 wherein step (d) includesthe steps of:(i) transforming each said section curve into acorresponding section curve in a predetermined planar coordinate systemto bring into coincidence said two points which are points ofintersection of the reference curve and each of said two sections and tobring into coincidence lines of intersection of the reference sectionand each of said two sections; (ii) computing an intermediate point bydividing internally into a ratio a:b each said corresponding sectioncurve transformed into said predetermined planar coordinate system, thedividing at the ratio a:b designating third and fourth points, anddividing internally into the ratio m:n a straight line connecting thethird and fourth points at which said section curves are internallydivided into the ratio a:b, the straight line dividing designating theintermediate point; (iii) computing an untransformed intermediatesection curve in said predetermined planar coordinate system bycomputing a number of intermediate points by changing the ratio a:b; and(iv) transforming the untransformed intermediate section curve in saidpredetermined planar coordinate system into the intermediate sectioncurve in said intermediate section.
 14. A method of machining a curvedsurface of a three-dimensional body on a workpiece using a digitalcomputer, comprising the steps of:(a) inputting, into said computer,section data relating to two sections of the body, and data relating toa reference section which contains a reference curve specifying theexternal form of the body and containing first and second points lyingon respective section curves in respective ones of said two sections;(b) specifying a plurality of given sections of the body in accordancewith said data; (c) dividing said reference curve into a plurality ofsegments at dividing points and producing dividing point data; (d)computing an equation for a transformation from a specific coordinatesystem to obtain an intermediate section which contains one of saiddividing points on the reference curve and which contains a straightline that lies perpendicular to the reference section; (e) computingdata relating to an intermediate section curve on the basis of datarelating to the section curves in respective ones of said two sectionsand on the basis of said dividing point data relating to the position ofthe one of said dividing points contained in said intermediate section;(f) computing intermediate section curve data relating to intermediatesection curves of a plurality of intermediate sections which containsaid dividing points and arranged radially around said straight line;(g) creating machining data relating to the curved surface of saidthree-dimensional body on the basis of a plurality of said intermediatesection curves; (h) generating blocks of control data based on saidmachining data; (i) producing a paper tape including said control data;and (j) controlling a machine to produce said curved surface of saidthree-dimensional body on the workpiece on the basis of the control dataon the paper tape.
 15. A method according to claim 14, wherein step (b)further comprises the steps of:(i) dividing said reference curveinternally into a ratio m:n; and (ii) generating said dividing points bychanging the ratio m:n.
 16. A method according to claim 15, wherein step(d) comprises the steps of:(i) transforming each said section curve intoa corresponding section curve in a predetermined planar coordinatesystem to bring into coincidence the first and second points and tobring into coincidence lines of intersection of said reference sectionand each of said two sections; (ii) computing an intermediate point bydividing internally into a ratio a:b each said corresponding sectioncurves transformed into said predetermined planar coordinate system, thedividing at the ratio a:b designating third and fourth points, anddividing internally into a ratio m:n a straight line connecting thethird and fourth points at which said section curves are internallydivided into the ratio a:b, the straight line dividing designating theintermediate point; (iii) computing an untransformed intermediatesection curve in said predetermined planar coordinate system bycomputing a number of intermediate points by changing the ratio a:b; and(iv) transforming the untransformed intermediate section curve in saidpredetermined planar coordinate system into the intermediate sectioncurve in said intermediate section.
 17. A method of machining a curvedsurface of a three-dimensional body on a workpiece using a digitalcomputer, comprising the steps of:(a) inputting, into said computer,section data relating to a section of the body, and data relating tofirst and second reference sections which contain first and secondreference curves specifying the external form of the body and containingfirst and second points, respectively, lying on a section curve in saidsection; (b) specifying a plurality of given sections of saidthree-dimensional body in accordance with said data; (c) dividing saidfirst reference curve into a plurality of segments at dividing points;(d) computing an equation for a transformation from a specificcoordinate system to obtain an intermediate section which contains a oneof the dividing points on said first reference curve and which lies inparallel with said section, and a point of intersection of saidintermediate section and said second reference curve; (e) computing datarelating to an intermediate section curve in said intermediate sectionon the basis of data relating to the section curve in said section andon the basis of data relating to the position of the one of the dividingpoints and the point of intersection contained in said intermediatesection; (f) computing intermediate section curve data relating to saidintermediate section curves of a plurality of intermediate sectionscontaining said dividing points; (g) creating machining data relating tothe curved surface of the three-dimensional body on the basis of aplurality of said intermediate section curves; (h) generating blocks ofcontrol data based on said machining data; (i) producing a paper tapeincluding said control data; and (j) controlling a machine to producesaid curved surface of said three-dimensional body on the workpiece onthe basis of the control data on the paper tape.
 18. A method accordingto claim 17, wherein step (b) comprises the steps of:(i) dividing saidfirst reference curve internally into a ratio of m:n; and (ii)generating said dividing points by changing the ratio m:n.
 19. A methodaccording to claim 18, wherein step (e) comprises the steps of:(i)transforming said section curve, said point of intersection and the oneof said dividing points into a corresponding section curve andcorresponding points in a predetermined coordinate system to bring intocoincidence the first point and the one of the dividing points of saidintermediate section and said first reference curve, and to bring intocoincidence the line of intersection of said section and said firstreference section and the line of intersection of said intermediatesection and said first reference section; (ii) computing a ratio L1:L2,where L1 is the length of a line segment connecting the first point andthe second point of the section curve transformed into the correspondingsection curve in said predetermined planar coordinate system, and whereL2 is the length of a line segment connecting the one of said dividingpoints and the point of intersection transformed into points in saidpredetermined planar coordinate system; (iii) finding a first curvepoint which divides said transformed corresponding section curveinternally into a ratio a:b; (iv) computing an angle defined by thesecond point, the first point and the point of intersection; (v)computing a second curve point obtained by rotating by the angle a thirdcurve point which divides a line segment defined by the first point andthe first curve point externally into the ratio L1:L1; (vi) generatingan untransformed intermediate section curve in said predetermined planarcoordinate system by finding a plurality of second curve points bychanging the ratio a:b; and (vii) transforming the untransformedintermediate section curve in said predetermined planar coordinatesystem into the intermediate section curve in said intermediate section.20. A method of machining a curved surface of a three-dimensional bodyon a workpiece using a digital computer, comprising the steps of:(a)inputting, into said computer, section data relating to a section of thebody, and data relating to first and second reference sections whichcontain first and second reference curves specifying the external formof the body and contain first and second points, respectively, lying ona section curve in said section; (b) specifying a plurality of givensections of said three-dimensional body in accordance with said data;(c) dividing said first reference curve into a plurality of segments atdividing points; (d) computing an equation for a transformation from aspecific coordinate system to obtain an intermediate section whichcontains one of the dividing points on said first reference curve andwhich is perpendicular to said first reference section and to said firstreference curve, and a point of intersection of said intermediatesection and said second reference curve; (e) computing data relating toan intermediate section curve in said intermediate section on the basisof data relating to said section curve in said section and on the basisof data relating to the positions of the one of said dividing points andsaid point of intersection; (f) computing intermediate section curvedata relating to intermediate section curves of a plurality ofintermediate sections which contain dividing points; (g) creatingmachining data relating to the curved surface of the three-dimensionalbody on the basis of said intermediate section curve data relating to aplurality of intermediate section curves; (h) generating blocks ofcontrol data based on said machining data; (i) producing a paper tapeincluding said control data; and (j) controlling a machine to producesaid curved surface of said three-dimensional body on the workpiece onthe basis of the control data on the paper tape.
 21. A method accordingto claim 20, wherein step (b) comprises the steps of:(i) dividing saidfirst reference curve internally into a ratio m:n; and (ii) generatingsaid dividing points by changing the ratio m:n.
 22. A method accordingto claim 21, wherein step (e) comprises the steps of:(i) transformingsaid section curve, said point of intersection and the one of saiddividing points into a corresponding curve and corresponding points in apredetermined coordinate system to bring into coincidence the firstpoint and the one of the dividing points of said intermediate sectionand said first reference curve, and to bring into coincidence the lineof intersection of said section and said first reference section and theline of intersection of said intermediate section and said firstreference section; (ii) computing a ratio L1:L2, where L1 is the lengthof a line segment connecting the first point and the second point of thesection curve transformed into the corresponding curve in saidpredetermined planar coordinate system, and where L2 is the length of aline segment connecting the one of said dividing points and the pointsof intersection transformed into points in said predetermined planarcoordinate system; (iii) finding a first curve point which divides saidtransformed corresponding section curve internally into a ratio a:b;(iv) computing an angle defined by the second point, the first point andthe point of intersection; (v) computing a second curve point obtainedby rotating by the angle a third curve point which divides a linesegment defined by the first point and the first curve point externallyinto the ratio L1:L2; (vi) generating the untransformed intermediatesection curve in said predetermined planar coordinate system by findinga plurality of second curve points by changing the ratio a:b; and (vii)transforming said untransformed intermediate section curve in saidpredetermined planar coordinate system into the intermediate sectioncurve in said intermediate section.
 23. A method of machining a curvedsurface of a three-dimensional body on a workpiece using a digitalcomputer, comprising the steps of:(a) inputting, into said computer,section data relating to a section of the body, and data relating tofirst and second reference sections which contain first and secondreference curves specifying the external form of the body and containfirst and second points, respectively, lying on a section curve in saidsection; (b) specifying a plurality of given sections of saidthree-dimensional body in accordance with said data; (c) dividing saidfirst reference curve into a plurality of segments at dividing points;(d) computing an equation for a transformation from a specificcoordinate system to obtain an intermediate section which contains theone of the dividing points on said first reference curve and whichcontains a straight line lying perpendicular to said first referencesection, and a point of intersection of said intermediate section andsaid second reference curve; (e) computing an intermediate section curvein said intermediate section on the basis of data relating to saidsection curve in said section and on the basis of information relatingto the positions of the one of said dividing points and said point ofintersection; (f) computing intermediate section curve data relating tointermediate section curves of a plurality of intermediate sectionscontaining said dividing points; (g) creating machining data relating tothe curved surface of the three-dimensional body on the basis of aplurality of intermediate section curves; and (h) generating blocks ofcontrol data based on said machining data; (i) producing a paper tapeincluding said control data; and (j) controlling a machine to producesaid curved surface of said three-dimensional body on the workpiece onthe basis of the control data on the paper tape.
 24. A method accordingto claiim 23, wherein step (b) comprises the steps of:(i) dividing saidfirst reference curve internally into a ratio m:n; and (ii) generatingsaid dividing points by changing the ratio m:n.
 25. A method accordingto claim 24, wherein step (b) comprises the steps of:(i) transformingsaid section curve, said point of intersection and the one of saiddividing points into a corresponding section curve and correspondingpoints in a predetermined planar coordinate system to bring intocoincidence the first point and the one of said dividing points of saidintermediate section and said first reference curve, and to bring intocoincidence the line of intersection of said section and a referencesection and the line of intersection of said intermediate section andsaid reference section; (ii) computing a ratio L1:L2, where L1 is thelength of a line segment connecting the first point and the second pointof the section curve transformed into the corresponding section curve insaid predetermined planar coordinate system, and where L2 is the lengthof a line segment connecting the one of said dividing points and saidpoint of intersection transformed into points in said predeterminedplanar coordinate system; (iii) finding a first curve point whichdivides said transformed corresponding section curve internally into aratio a:b; (iv) computing an angle defined by the second point, thefirst point and the point of intersection; (v) computing a second curvepoint obtained by rotating by the angle a third curve point whichdivides a line segment defined by the first point and the first curvepoint externally into the ratio L1:L2; (vi) generating an untransformedintermediate section curve in said predetermined planar coordinatesystem by finding a plurality of second curve points by changing theratio a:b; and (vii) transforming said untransformed intermediatesection curve in said predetermined planar coordinate system into theintermediate section curve in said intermediate section.
 26. A method ofmachining a curved surface of a three-dimensional body on a workpieceusing a digital computer, comprising the steps of:(a) inputting, intosaid computer, section data relating to a section of the body, and datarelating to first and second reference sections which contain first andsecond reference curves specifying the external form of the body andcontain first and second points, respectively, lying on a section curvein said section; (b) specifying a plurality of given sections of saidthree-dimensional body in accordance with said data; (c) dividing saidfirst reference curve into a plurality of segments at dividing points;(d) computing an equation for a transformation from a specificcoordinate system to obtain an intermediate section which contains a oneof said dividing points and a corresponding dividing point on said firstand second reference curves, respectively, and which contains a point ofintersection of said first reference section and a perpendicular fromthe corresponding point to said first reference section; (e) computingdata relating to an intermediate section curve on the basis of datarelating to said section curve in said section and on the basis of datarelating to the one of said dividing points and said correspondingdividing points; (f) computing intermediate section curve data relatingto intermediate section curves of a plurality of intermediate sectionscontaining said dividing points; (g) creating machining data relating tothe curved surface of the three-dimensional body on the basis of aplurality of said intermediate section curves; (h) generating blocks ofcontrol data based on said machining data; (i) producing a paper tapeincluding said control data; and (j) controlling a machine to producesaid curved surface of said three-dimensional body on the workpiece onthe basis of the control data on the paper tape.
 27. A method accordingto claim 26, wherein step (d) comprises the steps of:(i) transformingsaid section curve, the one of the dividing points and saidcorresponding dividing point into a corresponding section curve andcorresponding points in a predeetermined coordinate system to bring intocoincidence the first point and the one of the dividing points of saidintermediate section and said first reference curve, and to bring intocoincidence the line of intersection of said section and a referencesection and the line of intersection of said intermediate section andsaid reference section; (ii) computing a ratio L1:L2, where L1 is thelength of a line segment connecting the first point and second point ofthe corresponding section curve transformed into the curve in saidpredetermined planar coordinate system, and where L2 is the length of aline segment connecting the one of said dividing points and thecorresponding dividing point transformed into points in saidpredetermined planar coordinate system; (iii) finding a first curvepoint which divides said transformed corresponding section curveinternally into a ratio a:b; (iv) computing an angle defined by thesecond point, the first point and the corresponding dividing point; (v)computing a second curve point obtained by rotating by the angle a thirdcurve point which divides a line segment defined by the first point andthe first curve point externally into the ratio L1:L2; (vi) generatingan untransformed section curve in said predetermined planar coordinatesystem by finding a plurality of second curve points by changing theratio a:b; and (vii) transforming said untransformed intermediatesection curve in said predetermined planar coordinate system into theintermediate section curve in said intermediate section.
 28. A method ofmachining a curved surface of a three-dimensional body on the workpieceusing a digital computer, comprising the steps of:(a) inputting, intosaid computer, data relating to two mutually parallel sections of thebody, and data relating to first and second reference sections whichcontain first and second reference curves, respectively, specifying theexternal form of the body, said first reference curve connecting firstand second points, and said second reference curve connecting third andfourth points, wherein the first and third points lie on a section curvein one of said two mutually parallel sections and the second and fourthpoints lie on a section curve in the other of said two mutually parallelsections; (b) specifying a plurality of given sections of the body inaccordance with said data; (c) dividing said first reference curveinternally into a ratio m:n at a dividing point; (d) computing anequation for a transformation from a specific coordinate system toobtain an intermediate section which contains said dividing point lyingon said first reference curve, and which lies in parallel with saidsection, and a point of intersection of said intermediate section andsaid second reference curve; (e) computing data relating to anintermediate section curve on the basis of data relating to said sectioncurves in respective ones of said sections and on the basis of datarelating to the positions of the dividing point and said point ofintersection; (f) computing intermediate section curve data relating tointermediate section curves of a plurality of intermediate sectionscontaining said dividing points; (g) creating machining data relating tothe curved surface of the three-dimensional body on the basis of aplurality of intermediate section curves; (h) generating blocks ofcontrol data based on said machining data; (i) producing a paper tapeincluding said control data; and (j) controlling a machine to producesaid curved surface of said three-dimensional body on the workpiece onthe basis of the control data on the paper tape.
 29. A method accordingto claim 28, wherein step (e) comprises the steps of:(i) transformingeach of said section curves, said dividing point and said point ofintersection of the intermediate section and said first and secondreference curves into a corresponding section curve, and correspondingpoints in a predetermined planar coordinate system, to bring intocoincidence the first and second points of said first reference curveand each of said sections and said dividing point, and to bring intocoincidence lines of intersection of said first reference section andeach of said sections and a line of intersection of said intermediatesection and said first reference section; (ii) computing an intermediatepoint by internally dividing into a ratio a:b each of said correspondingsection curves transformed into said predetermined planar coordinatesystem, the dividing by the ratio a:b designating fifth and sixthpoints, and dividing internally into the ratio m:n a straight lineconnecting the fifth and sixth points at which said section curves areinternally divided into the ratio a:b, the straight line dividingdesignating the intermediate point; (iii) generating a temporaryintermediate section curve comprising intermediate points by finding anumber of intermediate points by changing the ratio a:b and containing atemporary intermediate section curve end point; (iv) computing a ratioL1:L2, where L1 is the length of a line segment connecting the firstpoint and the third point and where L2 is the length of a line segmentconnecting said dividing point and the point of intersection transformedinto points in said predetermined planar coordinate system; (v) findinga first curve point which divides said temporary intermediate sectioncurve into a ratio c:d; (vi) computing an angle defined by the temporaryintermediate section curve end point, the first point and the point ofintersection; (vii) computing a second curve point obtained by rotatingby the angle a third curve point which divides a line segment defined bythe first point and the first curve point externally into the ratioL1:L2; (viii) generating an untransformed intermediate section curve insaid predetermined planar coordinate system by finding a number ofsecond curve points by changing the dividing ratio c:d, and (ix)transforming said untransformed intermediate section curve in saidpredetermined planar coordination system into the intermediate sectioncurve in said intermediate section.
 30. A method of machining a curvedsurface of a three-dimensional body on a workpiece using a digitalcomputer, comprising the steps of:(a) inputting, into said computer,data relating to two sections of the body, and data relating to firstand second reference sections which contain first and second referencecurves, respectively, specifying the external form of the body, saidfirst reference curve connecting first and second points, said secondreference curve connecting third and fourth points, where the first andthird points lie on a section curve in one of said two sections and thesecond and fourth points lie on a section curve in the other of said twosections; (b) specifying a plurality of given sections of the body inaccordance with said data; (c) dividing said first reference curveinternally into a ratio m:n at a dividing point; (d) computing anequation for a transformation from a specific coordinate system toobtain an intermediate section which contains said dividing point lyingon said first reference curve and which is perpendicular to said firstreference section and to said first reference curve, and a point ofintersection of said intermediate section and said second referencecurve; (e) computing data relating to an intermediate section curve onthe basis of data relating to said section curves in respective ones ofsaid sections and on the basis of data relating to the positions of saiddividing point and said point of intersection; (f) computingintermediate section curve data relating to intermediate section curvesof a plurality of intermediate sections containing said dividing points;(g) creating machining data relating to the curved surface of thethree-dimensional body on the basis of a plurality of intermediatesection curves; (h) generating blocks of control data based on saidmachining data; (i) producing a paper tape including said control data;and (j) controlling a machine to produce said curved surface of saidthree-dimensional body on the workpiece on the basis of the control dataon the paper tape.
 31. A method according to claim 30, wherein step (e)comprises the steps of:(i) transforming each of said section curves,said dividing point and said point of intersection into a correspondingsection curve and corresponding points in a predetermined planarcoordinate system to bring into coincidence the first and third points,and said dividing point of said first reference curve, and to bring intocoincidence lines of intersection of said first reference section andeach of said sections and a line of intersection of said intermediatesection and said first reference section; (ii) computing an intermediatepoint by internally dividing into a ratio a:b each of said correspondingsection curves transformed into said predetermined planar coordinatesystem, the dividing at the ratio a:b designating fifth and sixthpoints, and dividing internally into a ratio m:n a straight lineconnecting the fifth and sixth points at which said section curves areinternally divided into the ratio a:b, the straight line dividingdesignating the intermediate point; (iii) generating a temporaryintermediate section curve comprising intermediate points by finding anumber of intermediate points by changing said internal dividing ratioa:b and containing a temporary intermediate section curve end point;(iv) computing a ratio L1:L2, where L1 is the length of a line segmentconnecting the first point and the third point and where L2 is thelength of a line segment connecting the one of said dividing points andsaid point of intersection transformed into points in said predeterminedplanar coordinate system; (v) finding a first curve point which dividessaid temporary intermediate section curve into a ratio c:d; (vi)computing an angle defined by the temporary intermediate section curveend point, the first point and said point of intersection; (vii)computing a second curve point obtained by rotating the angle a thirdcurve which divides a line segment defined by the first point and thefirst curve point externally into the ratio L1:L2; (viii) generating anuntransformed intermediate section curve in said predetermined planarcoordinate system by finding a number of second curve points by changingthe ratio c:d; and (ix) transforming said untransformed intermediatesection curve in said predetermined planar coordinate system into theintermediate section curve in said intermediate section.
 32. A method ofmachining a curved surface of a three-dimensional body on a workpieceusing a digital computer, comprising the steps of:(a) inputting, intosaid computer, data relating to two sections of the body, and datarelating to first and second reference sections which contain first andsecond reference curves, respectively, that specify the external form ofthe body, said first reference curve connecting first and second points,said second reference curve connecting third and fourth points, whereinthe first and third points lie on a section curve in one of said twosections and the second and fourth points lie on a section curve in theother of said two sections; (b) specifying a plurality of given sectionsof the body in accordance with said data; (c) dividing said firstreference curve internally into a ratio of m:n at a dividing point; (d)computing an equation for a transformation from a specific coordinatesystem to obtain an intermediate section which contains said dividingpoint lying on said first reference curve and which contains a straightline lying perpendicular to said first reference section, and a point ofintersection of said intermediate section and said second referencecurve; (e) computing data relating to an intermediate section curve onthe basis of data relating to said section curves in respective ones ofsaid sections and on the basis of data relating to the positions of saiddividing point and said point of intersection; (f) computingintermediate section curve data relating to intermediate section curvesof a plurality of intermediate sections containing said dividing points;(g) creating machining data relating to the curved surface of thethree-dimensional body on the basis of a plurality of intermediatesection curves; (h) generating blocks of control data based on saidmachining data; (i) producing a paper tape including said control data;and (j) controlling a machine to produce said curved surface of saidthree-dimensional body on the workpiece on the basis of the control dataon the paper tape.
 33. A method according to claim 32, wherein step (e)comprises the steps of:(i) transforming each of said section curves,said dividing point and said point of intersection into a correspondingsection curve, and corresponding points in a predetermined planarcoordinate system to bring into coincidence the first and third points,and said dividing point of said first reference curve, and to bring intocoincidence lines of intersection of said first reference section andeach of said sections and a line of intersection of said intermediatesection and said first reference section; (ii) computing an intermediatepoint by internally dividing into a ratio a:b each of said correspondingsection curves transformed into said predetermined planar coordinatesystem, the dividing at the ratio a:b designating fifth and sixthpoints, and dividing internally into a ratio of m:n a straight lineconnecting the fifth and sixth points at which said section curves areinternally divided into the ratio a:b, the straight line dividingdesignating the intermediate point; (iii) generating a temporaryintermediate section curve comprising intermediate points by finding anumber of intermediate points by changing the ratio a:b and containing atemporary intermediate section curve end point;(iv) computing a ratioL1:L2, where L1 is the length of a line segment connecting the firstpoint and the third point and where L2 is the length of a line segmentconnecting said dividing point and said point of intersectiontransformed into points in said predetermined planar coordinate system;(v) finding a first curve point which divides said temporaryintermediate section curve into a ratio c:d; (vi) computing an angledefined by said temporary intermediate section curve end point, thefirst point and said point of intersection; (vii) computing a secondcurve point obtained by rotating by the angle a third curve point whichdivides a line segment defined by the first point and the first curvepoint externally into the ratio L1:L2; (viii) generating anuntransformed intermediate section curve in said predetermined planarcoordinate system by finding a number of second curve points by changingthe ratio c:d; and (ix) transforming said untransformed intermediatesection curve in said predetermined planar coordinate system into theintermediate section curve in said intermediate section.
 34. A method ofmachining a curved surface of a three-dimensional body on a workpieceusing a digital computer, comprising the steps of:(a) inputting, intosaid computer, data relating to two sections of the body, and datarelating to first and second reference sections which contain first andsecond reference curves, respectively, that specify the external form ofthe body, said first reference curve connecting first and second points,said second reference curve connecting third and fourth points, wherethe first and third points lie on a section curve in one of said twosections and the second and fourth points lie on a section curve in theother of said two sections; (b) specifying a plurality of given sectionsof the body in accordance with said data; (c) dividing said first andsecond reference curves internally into a ratio m:n at first and seconddividing points; (d) computing an equation for a transformation from aspecific coordinate system to obtain an intermediate section whichcontains the first and second dividing points of said first and secondreference curves, respectively, and which contains a point ofintersection of said first reference section and a perpendicular fromthe second dividing point on said first reference section; (e) computingdata relating to an intermediate section curve on the basis of datarelating to said section curves in respective ones of said sections andon the basis of the ratio m:n; (f) computing intermediate section curvedata relating to an intermediate section curve in each of a plurality ofintermediate sections by the ratio m:n; (g) creating machining datarelating to the curved surface of the three-dimensional body on thebasis of a plurality of intermediate section curves; (h) generatingblocks of control data based on said machining data; (i) producing apaper tape including said control data; and (j) controlling a machine toproduce said curved surface of said three-dimensional body on theworkpiece on the basis of the control data on the paper tape.
 35. Amethod according to claim 34, wherein step (e), comprises the stepsof:(i) transforming each of said section curves and the first and seconddividing points of said first and second reference curves intocorresponding section curves and corresponding points in a predeterminedplanar coordinate system to bring into coincidence the first and secondpoints of said first reference curve and each of said sections and saidfirst dividing point of said first reference curve, and to bring intocoincidence lines of intersection of said first reference section andeach of said sections and a line of intersection of said intermediatesection and said first reference section; (ii) computing intermediatepoints by internally dividing into a ratio a:b each of saidcorresponding section curves transformed into said predetermined planarcoordinate system, the dividing at the ratio a:b designating fifth andsixth points for each of said section curves, and dividing internallyinto a ratio of m:n the straight line connecting the fifth and sixthpoints at which said section curves are internally divided into a ratioa:b, said straight line dividing designating an intermediate point foreach said curve; (iii) generating a temporary intermediate section curvefor each section comprising respective intermediate points by finding anumber of respective intermediate points by changing the ratio a:b andeach section curve having a temporary intermediate section curve endpoint; (iv) computing ratio L1:L2 for each intermediate section, whereL1 is the length of a line segment connecting the first point and thethird point, and where L2 is the length of a line segment connectingsaid first and second dividing points transformed into points in saidpredetermined planar coordinate system; (v) finding first a curve pointfor each intermediate section which divides said temporary intermediatesection curve into a ratio c:d; (vi) computing an angle for eachintermediate section defined by said temporary intermediate sectioncurve end point, the first point and said second dividing point; (vii)computing a second curve point for each intermediate section obtained byrotating by the angle a third curve point which divides a line segmentdefined by the first point and the first curve point externally into theratio L1:L2; (viii) generating untransformed intermediate section curvesfor each intermediate section in said predetermined planar coordinatesystem by finding a number of respective second curve points by changingthe ratio c:d; and (ix) transforming said untransformed intermediatesection curves in said predetermined planar coordinate system into theintermediate section curves in said intermediate sections.
 36. A methodof machining a curved surface of a three-dimensional body on a workpieceusing section curves in given sections that define the shape of the bodyat different section lines, the method using a computer and comprisingthe steps of:(a) inputting given section data defining the two givensections and given section curve data defining the two correspondinggiven section curves; (b) defining intermediate sections that insure amachined surface between the given sections; (c) generating intermediatesection curve data for each of the defined intermediate sections; (d)generating blocks of control data based on the intermediate sectioncurve data; (e) producing a paper tape including said control data; and(f) controlling a machine to produce said curved surface of saidthree-dimensional body on the workpiece on the basis of the control dataon the paper tape.
 37. A method according to claim 36, wherein step (d)comprises the steps of:(i) creating a numerical control tape based onthe intermediate section curve data; and (ii) machining the curvedsurfce of the body based on the numerical control tape.
 38. A methodaccording to claim 36, wherein step (b) comrpises the step of:(i)defining intermediate sections parallel with the given sections.
 39. Amethod according to claim 38, wherein the given section data includesdata defining the plane of the given section, and wherein step (i)comprises the steps of:(1) inputting reference section data for areference section which passes through the two given sections andreference curve data for a reference curve defining the external shapeof the body having end points corresponding to one of the end points ofeach of the given curves; (2) dividing the reference curve by a ratiom:n to define a dividing point, the dividing point lying in theintermediate section; (3) transforming the section data for one of thegiven sections into intermediate section data by translating the datadefining the plane, where the translated plane passes through thedividing point; and (4) repeating steps (2) and (3) a plurality of timesby varying the ratio m:n by an increment which insures the machinedsurface between the given sections.
 40. A method according to claim 36,wherein step (b) comprises the steps of:(i) inputting reference sectiondata for a reference section which passes through the given sections andreference curve data for a reference curve, where the reference curvedata defines the external shape of the body; and (ii) definingintermediate sections perpendicular to the reference curve andcontacting the reference curve.
 41. A method according to claim 40,wherein the given section data includes data defining the plane of thegiven section, and wherein said step (ii) comprises the steps of:(1)dividing the reference curve by a ratio m:n to define a dividing point,the dividing point lying in one of the intermediate sections; (2)defining a normal line passing through the reference curve and thedividing point; (3) transforming the section data for one of the givensections into intermediate section data by translating the data definingthe plane, where the translated plane passes through the normal and thereference section; and (4) repeating steps 1-3 plurality of times byvarying the ratio m:n by an increment which insures the machined surfacebetween the given sections.
 42. A method according to claim 36, whereinstep (b) comprises the steps of:(i) inputting reference section data fora reference section which passes through the given sections andreference curve data for a reference curve having end pointscorresponding to one of the end points of each of the given curves; (ii)defining an axis perpendicular to the reference section; and (iii)defining intermediate sections radiating from the axis.
 43. A methodaccording to claim 42, wherein the given section data includes datadefining the plane of the given section, and wherein step (iii)comprises the steps of:(1) dividing the reference curve by a ratio m:nto define a dividing point, the dividing point lying in the referencesection; (2) transforming the section data for one of the given sectionsinto intermediate section data by translating the data defining theplane, where the translated plane passes through the dividing point andthe axis; and (3) repeating steps (1) and (2) a plurality of times byvarying the ratio m:n by an increment which insures the machined surfacebetween given sections.
 44. A method according to claim 36, wherein step(b) comprises the steps of:(i) inputting reference section data for tworeference sections and reference curve data for two reference curves inthe reference sections having end points corresponding to one of the endpoints of each of the given curves; and (ii) defining intermediatesections perpendicular to one of the two reference sections andcontacting the corresponding section curve.
 45. A method according toclaim 44, wherein the given section data includes data defining theplane of the given section, and wherein step (ii) comprises the stepsof:(1) dividing the two reference curves by a ratio m:n to definerespective reference curve dividing points, the reference curve dividingpoints lying in the intermediate section; (2) defining a perpendicularline from one of the reference sections which passes through thedividing point of the other of the reference sections; (3) transformingthe section data for one of the given sections into intermediate sectiondata by translating the data defining the plane where the translatedplane passes through the perpendicular line and the two dividing points;and (4) repeating steps 1-3 a plurality of times by varying the ratio byan increment which insures the machined surface between the givensections.
 46. A method according to claims 45, wherein step (c)comprises the steps of:(I) transforming one of the given section curvesand reference curve dividing points into the same plane where one of thereference curve dividing points and one of the end points of the givencurve coincide; (II) determining a ratio L1:L2, where L1 is the lengthof a first straight line connecting the end points of the given sectioncurve and L2 is the length of a second straight line connecting thereference curve dividing points; (III) determining an angle between thefirst and second straight lines; (IV) dividing the given curve by aratio a:b to define a given curve dividing point; (V) determining atranslated and rotated point by rotating the given curve dividing pointby the angle and translating the given point based on the ratio L1:L2,the translated and rotated point lying on the intermediate sectioncurve; (VI) repeating steps (IV) and (V) a plurality of times bychanging the ratio a:b by a curve increment to create the intermediatesection curve; and (VIII) transforming the intermediate section curveinto intermediate section curve data.
 47. A method according to claim45, wherein step (c) comprises the steps of: (I) transforming the givensection curves and the reference curve dividing points into the sameplane where the corresponding end points of the given curves and thecorresponding reference curve dividing point coincide;(II) dividing thegiven curves by a ratio a:b to define respective given curve dividingpoints; (III) dividing a straight line connecting the given curvedividing points by the ratio m:n; (IV) repeating steps (II) and (III) aplurality of times by changing the ratio a:b by a curve increment tocreate a temporary intermediate curve having an end point; (V)determining a ratio L1:L2 where L1 is the length of a first straightline connecting the end points of the temporary intermediate curve andL2 is the length of a second straight line connecting the referencecurve dividing points; (VI) determining an angle between the first andsecond straight lines; (VII) dividing the temporary intermediate curveby a ratio c:d to define a temporary curve dividing point; (VIII)determining a translated and rotated point by rotating the temporarycurve dividing point by the angle and translating the temporary curvedividing point based on the ratio L1:L2; (IX) repeating steps (VII) and(VIII) a plurality of times by changing the ratio c:d by a final curveincrement to create the intermediate section curve; and (X) transformingthe intermediate section curve into intermediate section curve data. 48.A method according to claims 39, 41, 43 or 45, wherein step (c)comprises the steps of:(I) transforming the given sections and givensection curves onto the same plane where the end points coincide; (II)dividing the two given curves by a ratio a:b to define respective givencurve dividing points; (III) dividing a straight line connecting thegiven curve dividing points by the ratio m:n; (IV) repeating steps (II)and (III) a plurality of times by changing the ratio a:b by a curveincrement to create an intermediate section curve; and (V) transformingthe intermediate section curve into intermediate section curve data.